Equations Involving Absolute Values

Example 1. Solve `|2x-7|=3`.

If `|a|=3` then either a=3 or a=-3. This means that given equation is equivalent to the set of equations: `2x-7=3,2x-7=-3`.

From first equation we find that x=5, from second we find that x=2. Thus, there are 2 solutions: 2 and 5.

Example 2. Solve `|2x-8|=3x+1`.

We need to consider two cases: `2x-8>=0` and `2x-8<0`.

If `2x-8>=0` then `|2x-8|=2x-8` and given equation can be rewritten as `2x-8=3x+1`. From this we have that x=-9. However, when x=-9 inequality `2x-8>=0` doesn't hold (`2*(-9)-8=-26<0`), therefore, x=-9 is not root of the equation.

If `2x-8<0` then `|2x-8|=-(2x-8)` and given equation can be rewritten as `-(2x-8)=3x+1`. From this we have that `x=7/5` . When `x=7/5` inequality `2x-8<0` holds (`2*(7/5)-8=-26/5<0`), therefore, `x=7/5` is root of the equation.

Therefore, there is only one root: `x=7/5`.

Note, that equation of the form `|x-a|=b` can be also solved geometrically.